Different equations used in projectile motion to calculate time of flight, horizontal range, maximum height etc.
Let
O = Point of projection,
u = Velocity of projection, and
α = Angle of projection with the horizontal.
Consider a point P as the position of particle, after time t seconds with x and y as co-ordinates, as shown in Fig. 1.30.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJkAAAC9CAIAAAAvA0jNAAAgAElEQVR4nO1deVxUxx3/eduoIU3soTZeqRo0NpqgJGqKqfEIGk0AkVsONaiIVzzaJNWYVKMpohEEgRUhixwLcokIohE1HpHGtGoSa49PmrSNrbYkwMq1O/3jV8Zh3rFv31uWZd3vH7Bv3ryZefN9vzl/v98AccFZAJ1dABdsBheXzgMXl84DF5fOAxeXzgMXl84DF5fOAxeXzgMXl84DF5fOAxeXzgMXl84DF5fOAxeXzoOuyqXJZIqOju7sUjgWuiqXhBCdTtfZRXAsWM1lbm4uAHz11Vds4JYtW2xWIhfUQimXZrMZf6SmpgJATk4OvTV48GCALizfTgOruSSEPPjgg5S86upqAEhJSbF90VywEmrkKScnBwCOHDlCCAEAl1A6CFTSEBgYCABTp04FgGnTphFCTCaTTQvmgtVQyaVer0eJzMvLs22BXFAN9c0jDnlSU1NtWBoXtOAel+zQVAmWL18OAP/6179sXSQXVOIel9Z2eGPGjAGA3bt327pILqiEyja2paUFAHr06GHb0rigBZq4dM1GHArqyQCASZMm2bAoTgl2jaWj4RKsDofd6NTEZVVV1eeff46/7fkBOjiEVWGfylHPZWtrKwD4+PjYsDTOAZY5APj73/9O7EKnJrl0DX/kYTKZNm/ePGrUqG3btolGsC3B6pkIDQ1FLvG7c4GCY2j06NEhISFSkY8fP85eFhQUqBYP9Vw+/vjjyKW1C0b3G+S5mTVrls0yUvdYZWUlMCCusY80ioqKSkpK7JCRSi6///3vHz58GIns3r27bctkWxw9evSPf/wjvfzDH/5g5wJUVVWR9t96cXHx6dOn8feSJUvkHy8oKFCYkRq9grKyMgAICgqicpmbm6swHTugurr68uXL9DI4OJi2ctjHd3QBaF01NzeHhoZygYSQtLS0wsJC/M2pZBQXF9PfK1asGDFixPXr14Upi0INl1g7HBSm09G4fPkyAMTHx9OQZcuW0eItXbrUnlyqQFpaGv29fPnyAQMGoFgrgZoXGzFihJDLjIwMFUnZFjExMViYpKQkGshyyf52EDz44INStwIDA7mQFStWyCRl9Yt98MEH/fr127Rp08aNG7Hi3nnnnX79+j3zzDPWJqUF3Ld/+fLlJ554gn5Y7733Hr3VuVy2traWlpYKC0zByUBJScnq1avxd35+/okTJ+it3Nxcb29vmbyseDGuNBcvXgSAyMhI5Sl0EK5evTp27FipBp/lDwVXmEJWVtbJkyc7qHitra1EYmFPOL7V6XSsPj5tcpWMSNRzuXDhwk7sKWlhPv74Y/nOOzw8HADS09MJIePGjRMtcGtrq/11z8xmc2trq8LOFT8IeahnIiwsDAC6deumOgXtoO28DJcGg8Exh2ky4AhWyLdWLju3aubPn7969WosRt++fRcsWCBVJNTmlYETrHV0bS4pWLG7b22GuiqXrBiNHj0ai1FUVAQAycnJnVKkTkdX5ZK0KQ6eOnWKnVDqdLpNmzYRp2gzrUUX5hIRHR0NALjKyvLn4tIKOAKX586dA4ADBw4Ql0FLV+dy5MiRAHD16tVOLIPjoAtzuWTJEgDYt28fDbkP21UWXZjL4cOHP/HEE2yIi0uV6CwukTC0TLJz1g6OdtVh1XeNXIaHh+ODdpaJTu+qHRDQ2Ni4cuVKFU9ycmlPLnG747PPPrNbjl0CQAhpbGxU8WRntbE4D3n22WftnK/jQ2t/mZWVZcPSKMFjjz0GAKw6lgsIrVzitrhdrZkA3n//fbtl14WglctDhw7ZsDQW8eijjz711FP2zLELQSuX9lFaR7mPiopyta4y0Mrl9OnTib3a2BEjRjz99NN2yKiLQiuXVGdXNRR+ByiUN2/e1JidE0Mrl35+ftoLgVqH8nAJpUVo5RIdqGmERdeIuGLn6inloZVLO1iSnDx50rU4oARdYE6CysouobQINVzOnTuX2JFLAHD1lErQyXKZnJy8ceNGmQhUx84Fi+hkLltaWuQjAIBroUchHLq/dGKhPHjwoM3TVFNTBoPhm2++QeX/juMSFwdcy+jKoYbLt99+e/Pmzbh5aRWXR48eVR4Z/ZRYX7p7uN/Uf1RWltlsVsElKrIqwaVLlwCANdV3EDjy96H+w8c94Q5qY6kGswpQx0MqvjYlYG2VHQoquTSZTMprytpvecWKFdSsQAWwYBMmTNi0adOgQYMAIDMzU11SorChdyXbQiWX6NiwI7hEWx8twsSZ7d0/Gnv2kEurgCt2e/bsUZ0CVyq0nbZF0RwdDseldjECAH9/f/z9ySefuOTSAmzOJbbDK1eu1D6nBAHi4uJsUkgHh1YubXgq29mzZ20iQ5jI2LFj3d3dY2NjX331VQyfN2+e5jI6NLRyacOzZtAAT/ucEgBYT2T3D7RymZCQoDpvdnyLbnpsIuXs3LSzXJ93CrRyee3aNdV5c37JtcwpWQDjvsCJmRNCE5e9e/eWidPY2Pjkk08qSQ1NRLSIOAto7w9PFE7JsVa5lHGmo9PpFi1aZDGpixcvDh06dPz48YSQoqKiGzdumEymM2fOqCuYcjgfnVq5RD9zQiipKYyD/uowJC4u7gc/+AEhxFq128uXLyclJXl4eOh0uqSkpISEBLR93717940bN3bv3i2c57i4/D8ol1LnXyqvKUznP//5jzBc/sGdO3cmJiYKZ5MUM2bMEAb6+vomJCR8/fXXCovXhdCuvqZPnw4ABoNhy5Yt8o9RLv/6179qyR4VX9evXz9x4kQamJaW9u6774rGv3z5cnJy8uTJk2UopHurQ4cOlYkmCupRb86cOfhDr9cnJibu379/7969v/jFLxISEm7cuKHllTsOQAhhXaceO3aMi1FbW1tWVsYFUi71ej2GcIKYmJh44cIF4emYbDR0eTZlyhTS3jd1U1OTj48P55l2586dorW/cePG8PDwCxcuyL+nwWCIjIxcu3btkCFDRNPx9vbetm2bm5vbmDFjYmNjw8LCoqOj8VPj8Oijj8rn1VkAQsj06dN///vf4/W0adN++9vfikYtLy+Htmk45ZJ1yfLRRx+x8Xfs2AGCdpLlcu3atQDwxRdfBAUFCe8iampq4uPjhRUaExNz8eJF0XJabN7Dw8OHDRuGSxMU3bp1Ky4u/u9//yvzoF6vp5uXDtjdAiGEE7v6+vpf/vKX+Jst8dSpU99///1f//rXpI1L1nksHo6wd+9evMzJyRESyQI1B9BMTJgXgvPWLEohPqXae1ppaekjjzzCZjFv3rxbt25ZfNARuaytrWU9RWdmZi5dulQ0akZGxs6dO/G3kMuIiIh169adO3cOLxsbGy04egeA9soDXO1wRMbExFhsSCloMRSitLR04MCBer3+9ddfx+xsu31tH/Cic+zYsdu3b9PLlpaWrVu3ZmRkxMfHJycnV1dX33sSgJW8ffv2nT17lk0qICCAjc8CV+ymTp2Kl5yvc65rdHd3/93vfmfVW2H7qSQm/YBWrVqVl5e3fft2awXUcQBr1qxhr7dv3+7m5kZHp7W1tQUFBXV1ddwRYvn5+RyXFEoaH3z2T3/6EyEkKirK29ubss6K49ixY61lkU1fxSNCsALqgO0qCyCE1NXVcQI0evRo+ceMRiMAvPLKK3hZV1f34YcfEsHbYiAHqRU7zgv+a6+9ZtWbsACAF198UXn8Cxcu0HxxR7Z///405OWXX1ZdEnvi/16Qm5qaCCF1dXX4OQcEBFh+EqBPnz5cIOVSr9d369ZNdN8Dj6rhAmUOpVABaxOJjY2lWfv6+nLpILC9dWTRhIqKCnrR1NSELFKB27NnD+3MPDw8tm/fjr91Oh1bX56enu+88w6X9N69e9988016ibWAOnbsTIa0rzKbeNNSwuWMGTPQAzQRNLDsqhA95nPWrFkOTicQsWG9oidl60sqNe4pbpiDG9EoGVIrvQrLpnwr9Nq1axyX3MLkkSNH6K1vv/2WOCqd97i0Cnq9XkVjiD0lFUqUUcSGDRvYmBorCwCys7OVRxaCK0OXoFNlz6RON2fw4MHUVJ1K5NixYz/55BN1xZCC8rLh6UdvvPEGHikEAGPGjAEA4eJXcXExXeB1TDpVcjlr1ixrucS1TVwCpCMdDw8PdQWQx8CBA60992/dunXAnIchuul29OhRR5ZO9SNGa7kEgMmTJ5O2Q7U6jkgVZSNtDX5iYqJ8NLax/e6774gj0amSy4MHD1pVX6+++ipGpicb0uljR9SFVf0lAs09hRs7QjgsnXaSSwAoKipKTk7Gp9hBZgdxaa1cZmdnL1y4UHRxQ4iSkhLadzrOOp9KLnFXROEhezh8pXB3d2fPc+4IQPsdmI4A1gC0X2DS+F1qPGJFJZdZWVn024+Pjx86dKhM5J/+9Kcsl+pytAoAsGDBgo7OpbCwEN+I5tW5ja3KmsVDtJUQU1JSwhJpEw1Yi1AxLqOWZaNGjVL+bL9+/bR/o+hYTPXjFCqToCtbFmM+/fTT9G21KEZbBRVc0hWMQYMGDRs2TPmzAwYMYMdBKtDY2GgTV3QquVToOB/PlLYzkUQVlzgbGTNmjMU9Iqns7NN9yBVD3WMKuaQvKaV6qQ69evWiGg4y+SpPEAD0ev3NmzfBGjNe9lQWruNUAZPJpHXlUt1jSrhEY0q2m7TJ0KBPnz7Hjh1jt3eEsJZLNze3kJCQadOmWfUU+zp0v7MjnDApRAdy2UGtK261yuCjjz5CEVGu9YMLeAAgpZerBPR96YzT3mcpqXvMIpevvfbak08+CW3rduq21Sxizpw5NTU1+JsmjvrKVhlX0x0bLYWh60Fz5szRko4ULNZeh3DJDnmUlOPtt9/mQiyWm34c1DsiZwFolXugTZs2aRRKBGpvd1BLiy+4atUqqszMAQghzc3N3t7eXl5eytOV4RIVX4V3zWazUPeAEFJfX9+zZ0/RXMrLy6VWiDjmuLvQtuPx1ltvyb7Hvfi2GoXSjtMmqQkBAFR7mb+F/4qKioSGBjJgDw0SZob44osvMMRsNl+6dOnu3bu9evVSmD5SFRAQUFlZKRNB+JuWIScnJyIiol+/fvIZXb161dPTU6FQGgyG06dPW4yGrx8aGhobG3v+/HmL8aVQXl7ODtdffvnlVatWEUJYfeZ2+ZrN5u9973t4gZXS2Ni4dOnS0NBQqTzOnDmD6z4cl6dPn6ZEPvfcc6gyef369UOHDln8UO7evcuZMBCBwAUHB7MhOTk53FmmCPQQlJGRUVBQwIaLrnZiaWNjY7lwrAo/P78rV67QQKPRyJ5jl5ubGxMTwz61cOFC0maFCIyxjUVIaQ2Kyvfdu3fFI+O//Pz8U6dOKcyYELJ48WIQOIRBD3nQZu7j7e2NpaECKgOj0cgWwGQyCbXU9+/fz4WIOr7E74yauRNGcOvq6qTcSB85coRVb8BHUlJS8IMQrRyj0cherl27lg7EfvKTn2BVxMfHi2ZnMpk2b97MhoCyg69lRhJACElISOD4NxgMH3/8sUyKQkfA1dXVot2kxRIgjEbjhAkT8HdJScnVq1exp+T6xU8//ZRezps3b//+/cIPhX5nV65cOXv2LGdiLdWW3rlzZ+3atVIF9vLykjJFopHT0tKo5RC2DSBrNw4AbLsiOvoNDg4eMGCAMC/RQv6/3h966CHWktLi9rqQS24zBNuiwsJCJbu7iJEjR+KiqE6n6927t+g3wS7NHzhwAFelqTQgUHE5IyOjX79+QnkCACkDSnkDGJlhFNapv7//66+/TgPpUgn2NWxMiscff1wmR4sQ53LXrl1SowwWFRUV2BBxXEZHR2O5MzIyqHnC7du379y5w+lD//nPf8YfnPEJOqRliee4rKio6N27N6twKxqNtMllbm4ujhQUYs+ePQDw1VdfsYHUmFCv18+YMYN7pKqqCk0kRFsdOj9hueQQGBiopGzs7FxG1VTk2+fGCF5eXlu2bPn666+NRmO3bt2QA45LrnU9dOgQtD/mFO0LqKbP0aNHN2zYAO1PHTl8+DBXEnbAJtXhC4H6KDk5OdzYh+LIkSPCQCyYqO2pVAfh5+e3ceNGamnz4YcfUl1wBJ2ffPPNN1xSuNHERmanpBiNTuEUrrQAIaSlpYVNl3smMzOzoaHhkUceYQORyw8++IAwRLImIlzHCW0LQPg7MjLy5z//ORshLy/v3//+N5vFiRMnWltbCfNtccPmysrKmpqa3/zmN9wrIZeenp5sYFNTEyrgs2OfmTNn0rkaFpjrEbi+ioNOp2O/18zMTGEjgcmyLmfLysrWrVtHCBE9blJ+3lxRUSGzAQyEkPT0dDruUAjksrKyEj3oA8Dzzz9P71LvsnRlkhNBvEsVGAkhtbW1XNHpqFXqe/Ty8qqoqJg9ezYXjlwWFhYGBgbSNzebzSjZyFZpaSm3Wotb0Jz7hZCQEC5xlMLBgwdzsxFCiOgeJN2svn37dnFxcY8ePQghotbXffv2FX1NFocPHxYdRiCAENLa2hoTEyOc3skAuVy5cqWXl5fogA3nWGgmfezYMa4E+Ahdi7Ltai1yiephUket0sEdAAQEBGAdAUBERAQrXsI6wS/MbDYrd9CJ8xOLvjIbGhoUJigF8TmJRdA1PK6npMABCLIlTB8A5s+fTy+FLCrRYiotLRVd4UMu2fmlDHDAzIFOFaytFgR9HXwLdMugxPDPNvuX1mqAUR0RTsIoQkJCuP6SCu6qVasAABebpE5cU/JWcXFxxcXFwnCWSyUSjzaELGy1SYf50gGtxZmeRmjS95k/fz4APPfcc8IIKJfYI+J8Iy8vjxASFxdHqwwNGbSUXhTYvCuUS0II+tfHkoi2MdqByb700kuqU1Bkba4uaWxj+/TpAxKnGaIaOF3MPHz4MB0F4KF56enpAKDQ+aFVQM1V5VxSuQGApqYmXFNV/qySaPTd8dImIwORHQWLMURBnRj5+PiIRkDrHxVF1I41a9ZYxSVpb7ZnVV4WByycQpANnRRbwWVTU5NwUAAAQUFB7OUDDzxgMBji4uIWLFhw6NChlJQUvV7/zDPP4F2DwZDShtTU1LS0NJ1Ol5qaiiGJiYk2dAtNgYtnVpniUoOCjjv9ltbY3/72N9EIW7Zs4czFOaD8sI4Dt2/fzi6bS36GjY2Nolw6DkpKSjw8PHx9fQHg+eef9/HxycjIyMrKwlWI4ODgzMzMFFkcOHAgPT391q1b1AvismXLuLretWsX/a1kw0cKuM4F0m5ycQjy0ksvffnll6IRaF+Al2+99RYAVFVV0QhydtGtra3+/v5L2mPp0qURERGYqJeX1/LlyzF85cqVQ4cOffjhhydNmkRHE84NquBTXFw8ZcoUmU9Hr9fPnDmTPpiXl7dv3z760eAS94EDB/Du3LlzASA3N1c4EMHtPFRlAoAhQ4awd1XauKO8GwwGizHLysqUbMs1NTWxbhVxbQx3sPV6fWhoaEhIyLlz50JCQpYsWRIVFbVEAjExMXSOsXnz5h/96EcAgBur9C9OilhMmjRp2LBhHUi7MuC8gENqaio7je7Zs6efn9/s2bMHDRrE1SHfxiYnJ0+aNIkmxG6jU1h1ANR7771XXl5uMZoNgRth7AKhFpjN5oqKiuzs7JMnTwYHB0dHR5eXl4eGhkp9TIjIyMhRo0YNGTIEtxOeeuop4SxWiJ49ewoD+/fvzzkPxXDhgRHtuGSN6372s5/hj9raWu4Zq7hUUXcaU8BxrPw4wjHh7u7Osijl/w/vCls7kWVSeoktuFDxp+MOgLIJlixZAgCff/55ZxfEOnz55ZeUxREjRgh30fEr37ZtG8ZBhwks7jGHq27ctAxnIP/4xz/YQAfnEvUzHKRsyg9IoSzKpHPjxg3sQYcPH049Q95Lgf7itNwQSDDXNON+J9ja4ocQ4uXlpVCjVQa4HsvOJRwEO3bsEN19xFXl2bNnW1TtmDBhAgDEx8e/+eabILASvEceDvO4h5FLoW5gt27dQGxJXSNwiT8vL0+LWim2sbYrlCJYXMR/44032F5cxbBgypQpADBq1Ci85DpEwnLZt29fYRWgsCo8Y8tWG5D19fWiGiEKtzkVcvmrX/2K1epjs1ANOYXHNgijubu7K/l28XHqhBUVUFhTgHvvHBgYCAJNEex7/vnPf3Lp9u/fXziTtbnpD5egfPolJSXoP1whlx4eHsKDAizmohosl1xG8scvUeTk5HCboHPmzGFf4V7SSUlJAMA56BYKMqKhoaGlpUVKw99oNMpok1oETpmlUn7ooYdEb925cwe3SHFxRGrZUx60BVOB/Pz89evXi9767LPPAAAPJcddTNYO183NTXWmLIB1nIzM5eXloZYUTjGpzpUQdXV1wtEUIaS2tlb0kyeENDc3A8DNmzdpSFBQ0KlTpzgVLOX7CUI9dGxLrHW5TggJCAjQ0tF+9913Um4HRo8ejbt7P/zhD4U6zZz6SHFxMetmKDs7G82tFenhAWPYDe0h7yXHbDb7+fmJ3kJHlvj77t278ocXeHp6surkMofec+/j7e3NLi4jVI99mpqacMOcRXNzc3Z2ttAcR+rQMSzhsWPHWBkFgPDw8KKionnz5gHAtWvXxo0bh7fQCIdNoaGhgbNuYAEAePaZyK0pU6ZwNg9paWkhISFhYWGrV68uKyvz9/cXrRockgmncTU1NXFxcX/5y19oSF1d3fnz58PCwnCsYTAYLl26RJh2pr6+nq0sTFl+ZrZ161apc2ZRv0b0lhIo7y85HU9cIMXHjUYjPYLg008/5SSEEEI1p41GI6vHKzNEwN+tra319fWi5QHUQZWH0LMYbtCgEh4NTElJmTp1KhaXE8R9+/ZRHXBCiOgy7/nz59m2l0hU64YNG65du4aTIlFtZqk+nkNVVRV+UhTZ2dnUXkc0axpYWFjIDf1EffRg/PXr1wPAmjVrVqxYQffw2W1I4aSTzb2yshLXgKyzi75165ZQacNsNgtdBLCD3oSEBOroDiF02ElVxVH3h4ajrglpO3B9zJgxpL0GtxC4aJ6fn08NH4KCglgR+fGPf6yEy7lz5wrV+E6cOKHQvxsA4LCZgjoRKS0tpYFXrlyh3xYqW+Mlu4c8d+5c0d598eLF48aNy8nJUaq37unpGR0dTa/ZpQSj0UjNPzggB927dwcxdO/e3WAwpLXh4MGDWVlZOp0uISGBk7xXXnkFAHBDlO0Gzpw5s3///rT2SExMvHXrVnp6OpJN/bhRZGVlNTQ0cBrxUkhKSuK+ucrKSj8/PylLVUR5eTkdLUq1h3fv3qVWNOjVYvjw4TQaviy7LLVr166tW7cK8xLOBXbs2CFTNpDRWPn222+lKkWoH6sQQu85GI4mmxQvvviiuvQpZN6ZCHRI6+vrLRoSYeHz8vIAQGZqzxKMntyF5zA9/PDD7KWUye3MmTOpZ1Ycl8i/l6QZCV5Szw4c6OItChYH1Bb38/OLbI9FixYJx37oboQ1zEDgUXtBQUEVFRVnz57Nzs4ODg4mhGRmZkZFRXEmghTUJkLmnaUaKx8fH+GqCAvce/f395eJwwIAsF3lzNNiYmLY5c+oqCjhxLSqqop1EoA/5Ac3ku8s3zoLF+J9fHxohXLHHcoAu0lo055VDhxQUIwfP37ZsmV4C89asyo1CiUPqvARwnmwTk1NZY0jRO1tExMThYZv8lD5zqKbKjqdDsdECqsSXcd4enqifoZVBZg4cSJmFBMTExYWxt6KiYlRx+Xx48dFzflsCCUTHtWLiCq5xLUV0bxTUlKwMbQIVMYhhFRUVED7E17k4efnhyyK3lU9v+w4GwFaRQ7HJdURIWKTWYUwmUxeXl7oYsVoNL7wwgu43NO9e3f50ZowI97UW3HDIERGRobUSpbjQ+Val5b6kkddXZ3CmFKfDi4j2K5EXQYq37m2tlbUH4uDQKPj8i4KTfaFHbTV54I6aOLSAeHIZeto3I/9irPCxaXzwMWl88DFpfPAxaXzwMWl88DFpfPAxaXzwMWl88BOXCpfjrmfF240wuHk0sWlatiPy169eknpWS1evJgwZ0q7oA72qDsqaqhIAADR0dEhISGhoaGoa4IU4r4jexiBTFIy2LlzJzqWlv8y0F4H4efnR9XMaRaTJ0/Gu76+vuin2MHbDLtyiYZOwiqm5wYpUaKXr9DMzMwXXngBc5FxokUP8GLBRsADaJYvX04PG7lz547FsnUu7NqmidaaRXAWr0qEA6mSMWzy9fVl/dZjqdBzMWlTUJ44cSJeop9GLV4m7QO7cjl+/HgAYNXm0HcdIWTr1q1YodTNPGl/PvHixYsPHjwo5V+MA9r1yxzbwykeYBZUgxk9HFG7QdRvdvy+vHPkcvbs2QCAFmRYR++++y7+phb9KFtTp049fvz4kCFD8K7CA4DR/luhFhY652B9O2FerFkkhnS0M1+NsB+XaBVEKYmMjES3hPR7Rz1NakCIsoWCSL2eKcwLtdctOp28ePEiFX2qBxobG8uJKRE0wo4Ju8olGlqz55ChHz/8jerL90oGAG0u91BGlbiExd4UpzdSRzFSJCUl4cGnCLSDoGZ1bEwMsep8VPuj88c+zc3N7F2q4Y+XERERpK3jVH4qN6quK5fjBx54gMYXPaPWxaUgMwAZ8UJLBPY8WWAgejwikRjWik5+5AfAND66NHIQLtF7k8J5bSdwmZ+fT9o70mDvUl9e169fBwAvL681a9bIeLoUfU80sxV1byilaM/yh7/ZkQ6GcEc82QFoDutwXFKzfezGpLikrjjwEo0Xs7KyUlJS6HQF+10Z+zp8Njs7mw2sqakROiJl/aFTfxBubm7AeNTDr0p5i91ZsFP5qDtEhFBiWHvK/5dMDOwt4UleCDwHDsH678KJCrbhOM6i7jqwP6Y2kQUFBfg4mn3hpzNy5EibVUfHwH7fWlNTk8lkMplMLS0tQhuBHTt2VFdXZ2Zm4qSwpqYGAJ599tmoqKjQ0FBqhk1NFemISYhZs87/p70AAACJSURBVGaFhYXRtTcajkemoPOZHj16sJ/IwIEDOR+feBIUxWOPPWareug4OGi7gV5xOZMVqS5QCs3NzZzXDDodys3N9ff3DwkJ8fHxkXIrlZ6evmjRIh8fHzv7pVYNx+KSO80DAKifILTSRfcbDr5f0VlwUC6xCWWHKi0tLeg8wUWkFByLSxe04H8+AmGQD9MaiwAAAABJRU5ErkJggg==)
The equation of the path of a projectile or the equation of trajectory is given by
![Equation of the path of a projectile or the equation of trajectory Equation of the path of a projectile or the equation of trajectory](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPgAAABGCAIAAABuc85ZAAARCElEQVR4nO2de1QVRRzHfyoYEEcBC1BAU3yUookv1Aopk4MnS0FE0pQLHMBnqYAIJomWASkSoECaGRCSx/JAR8mwB2l4uoAhJiAobwQMkRQJuMD0x687Z7vPvU8usJ+/dmdnZ367+53Hzs78FgghPT09hBBCSF9fH26cOnWKMKDhssFoJSUlDx48YBOfENLR0cEyJktYmqppdMQMTVBRUXH79m21JMXyLtXW1paXl6uYFxBCfv7558bGRhkZKyT06OjogoICFc1ShdjY2NbW1n40gGJjY9PU1NTfVqgZgUAgEAhUSeHTTz+tra0lrHV169YtPp+vSo4EhY6I58rn8+/fv69iBjLQRLV369atJUuWtLS0qD1lDnU9Lz6fX1hYqMYE2QAyMuvq6urt7dVc3kpc5+XLl+XGqa2t1VA9WllZmZGRoYmUOTQNiAfFx8er0nUuKChobm5WwSRZ+Pv7P3z4UHac7u5uuW2rSBkrKSm5d++e3Nxv374NAE+ePJEbUxpnzpz58ccflT5d+/T19Q2O9w0JQgeAmpoapVNUoh3Iz88/ePAgm5ipqamqd9eImNCvXLlSUVHBMrIqrVxBQYG63uS0A157Q0PD0aNH+9sWlZAgdKXRkaLP0owbN27U19dLPFRcXFxXV6dogoOYGzdu5OXl9bcVKqFOoesILHXZ3d1Nx1WlHRIZZuVAsrOzmbsDoiIYhELn0DTHjx8XD3zy5Inav4qoEU7oUtHoiNPgY8SIESAkKiqqv80RhRO6VAZEi6xNpN2Qjo4OAHB0dAwICNi2bZuhoeGUKVO0bJtcOKFzsEWa0GtqaiZMmEB3FyxYAAC//vqrlsxiByd0DjUzY8aMyZMn97cVonBC1y0A4Nq1a/1thUrMnj3bzMysv60QhRO6bjHQ34BjYmIAQHOfxpWGEzqHOtFNlRNO6BxqxMTEpK2tjRDS2tqqa/OTVRU6j8djv8yCYxAzYsSIwMDA9PT0U6dObd261dPTs78t+h9cjT7kOCJEjWlaWFjA/8nPz1dj+qqjqtC5ryoDiLi4OFShn58fVaSNjQ2bc6mUTUxMMCQgIAAAJk6cqEmT1QZXow8V8vPzUanXr18nhEyfPh0A9PT0FHp3nDRpEgC8/fbbALBo0SIejxceHq4xk9UJJ/ShAgp97ty5uJuQkAAA06ZNI4o0y0VFRVhatm7dqilDNQMn9KHC1atXAWDx4sWEkNraWtTr77//rlAiVVVVeGJpaalmzNQUA0zohYWFERER/W3FQGXfvn3M98UtW7YomsL8+fPxXO2vblaRASZ0DqXp7Oy0trbGHvY777yDDicURV9fH4X+559/qt1CjTLwhK6i85khC67sBgDZy3MFAsGRI0e++eYbInZLASAkJOSll14CgKSkJEKIRr2hqJeBJ3QZ6NrYrU4xatQoZr8lMTFRYrTJkycDwKRJk4hQ6LNmzaJnEUKCg4PprpeXlxavQCUUEHp6evqxY8eY3hpSUlJycnLYnKuQ1wTl6uauri4AqKysVOJchTIdiE2HkZERAGzfvj0gIMDFxUXGCyUOsY8fP56GWFpaYnx0NFJcXIy7vr6+ippx5coVLC3ah22utBBbWFhgyPnz5wFAExMyqZK++OIL9vH37t0LAF1dXSwTZxku8dDA0npERAQAlJSU0JCbN28CQHR0tMT4QUFBd+7c0ZAx0hakaxoFildFRQVqHefrlJeXA4CRkVF3dzcRe/bffvstiDFjxoz09PTTp0+vX7/+9OnTsswSsmLFCgwZOXIkACQkJEiMv3PnTnrK7t270cdQR0dHXFxcbGxsTEwMbYionfjdhBCC85BkMNCF7uPjAwB3796lIfHx8SJVskLuNQciirUj7u7uAPDo0SPcNTU1NTAwkDiFuqioyNPTc/HixeJyp3R2dkrLCNMcN24cACQnJ9NTbt68KTF+cHAwloTRo0cDQHt7e0dHh56eHgAsXbp04cKFAJCeno6Rd+3ahaklJCQ8/fTTuG1tbS338gUCQWRkpNxougZ2Nvbv35+cnNzc3JyammpsbAxKuRwbKkLPzs4GgMzMTNwdOXKktOZPLaDQAeD999+XG3nKlCnAmAydlZUFAPPmzSOE5ObmYhnA5xQWFvbUU09hyqtWrXrttddw++TJk0T6szx06BBGGzt2LCGkt7d3+PDh9vb2cg3Tjjju3r0rzb9aXV2deI0TGxureqYDSPf/CZ2lxdXV1QBgaWlJCMnLywPhC4qGGDt2LAC89dZbcmNmZmbi86POt9BU6v9twoQJtC0SCAQGBgYAUFVVhUdNTU0BwMHBQVr6f/zxBwCsWbNm2bJlAEAIiY6OBoDAwEA2F6I5QdC3fD6fX1xcrLV8KWr08VRRUSHDMaCKKFaj19TUoJ6ampoAYP78+dJiZmRkyOi0TJs2jc1wjZWVFcavrq6WG3nixIkAIP4d5Pz584aGhgCwcuVKkZRpZPw87uTkJC1xGxsbEA4XbNy4Ea2aOXOmXKsQdQnu3LlzaklHN6mrq8OKA2/Xnj17mC/QKvLfw8vLy/v444/ZnIDddES209p33313w4YNXgz8/f3Nzc2dnJwAICgoSL5xAADg4uIiNyZ2VJjabWtr27JlCwC8/vrrWA0DY/ARR4tp+cFxCSp0cV2OGTOGdtJu3LiBqbF5DPTFgD3Nzc1xcXEnJZGWlnby5MmkpKTvv/+eEIKdbFR/VlbWTz/9dPny5eTk5Li4OEJIV1cXjhMQQr7++mtFzRhkKDyomZOTg49ZFWf7bJYAA4CHh8fIkSNxTLejo+PMmTMy4uP3bSr0sLAwANi3bx8R9rJMTU3xwV+6dEmkoTA3N5dRo9fV1QHAsmXLcPfAgQOyq3+E5iIDie/WP/zwg9wTtcPq1asBwNraOiMjAwtYaGioiYkJ7jI5ceKEeLHE8iZCXFycxHBNo7DQf/nlFwD46KOPNGFNS0sL9Wy2d+9e8v/vee+9956Mc1HodCQBT6FHsVOOg7hNTU2YCxU69uBlaNfY2NjR0ZEZ2crKSu7l0PIcGhrq7u6enZ2dnZ29du1abN9cXV3ZOGUPDQ09cOAAboeHh2/cuNHHx8fb29vLy4vH42Gr5ezs7Ofn5+3t7e3t7ePj4+PjwxOCeXl7e9NtCoZs3ryZ3uEdO3aYmZn5+/t7eXn5+vo6ODjY2trSo9h/0xpmZmZ02gLywQcfyL1d0lBY6Hp6enT+oNrfdcrKyuhVYQgdj6cDL9JGtZ999lkAcHBwwFdYaaSkpFRWVuI2HSfF/v3y5culGYbt2OHDh0NDQ0USDAkJEY8/gIYj1MKePXuqq6vv3LlTVlZWXV198eJFNze33NxcT09PkaJFy6GXJHx8fJydnQEgMDBQ/NmtXr1aaQsVE3pQUBAwqkkNPU6Rb0l+fn500FDGVAJXV1dmIfHw8KC769ato9uRkZHYQQcAY2Pj3t7e9vZ2elTaF8H6+nrmHSfCcgVSvsWq685otMDInpcxyMqqfKHX19fPmTPn7Nmz2Mb174Q12Xc/KCjI1dX177//xt3g4OB169Z98sknhJCysrLly5ej8UFBQRs3btywYQN2CYqLi11dXX18fNzc3GRcXXl5uYeHh7u7O3o9KC0tXbVqlXqXGGuUs2fPigdquiBpOUcZyBc6zotAkpOTtWCTEgyy6kftmJiY4BM0NzfXWqb4UHp6ephlTHeFTggpKipycXHBL/a6KSlqFW6I7NJtZrjIIfFzxbOgiKegs3R2duKLeFBQ0LBhwwDgxRdf1KYB+BEDAExNTbWZrwiyhK5TTQ97xBVMt+XKVPbVyShFuga1DUfJ8AM2ft9lM21BLeACU/Sb/sYbb4BwLXa/oLDQdR9VzB6glyyD4uLisrIy3C4sLAQAOzs77WRtZmbm4eFBd/X19adOnaqdrMUZVCuMOGRz9+5dANi8ebN2smN+OMeBY1tbW+1kLQ4ndF0nISHhq6++euWVV7788ssLFy6wPIs2TcyP0EZGRuPGjVO/iSzAWVI8Hq9fciec0HWZo0ePinwxMTIyUjQRqvgXXnihH0eHp06dun79+n7JGuGErltQXZaUlKC4DQ0Nhw8fTrUeHx+vRLLodRGn+mRkZGhT7g8fPvT19bWzs/vrr7+0lqk4nNB1FPSNSF1yNzQ0oNBxKYlEJL5Jt7W14fdsW1vbzMzMc+fOAcBvv/0mIwVpw1bKIRAInJ2dcWz6woULqamp0mzG7R07dnz33XdqNADhhK6L3Lt3DwBEfnnF5/NR6+ILLGSAU+1FYF+5qqizEydOAEBMTMzZs2cPHjwIACJ+08XTb2xslPYTAVWM4YSuu+B7JH266OBTT09P5M8LZ86c+eyzzxISEv755x96YnR0dGtrKyGkqqrKzc0NZy/yeDx3d3e6CEsi0dHRiYmJIj9BJ8K57wqVsYKCAvEyJs33zueff07jWFpaXrp0iRCSlZWVmJiolkVMnNB1HSr0I0eOAMD06dNFIvB4PNTHtm3bCCHt7e244pu6OWBJYmIiALz88stUcMyFbcz3BAsLC5HfiI4ZM4YeNTMzo8ve6ZgPTtSTWCV3dnbi4gEnJ6fIyEg0Axfm2tvbA8DFixcVuhCJcELXXURkgXPoxXsdx48fBwB9fX06Fx+FrpDfsqSkJGC4HR0/fjwAPP/880ToYNrPzw8PNTY2oqCpr6958+aBcCHO1q1bAeDq1avss0bfGwDQ3t6OIW1tbQDg5eWF4bm5uexTkwYndN2FKXQ7OzsA2LVrl3g0XBtBHZ/Pnj2bucuG69evo6ToiqfKysqVK1fu37+fCL180WXmhOGs69atW3S7oaGBEPL48WNgLFFnA5YTOjsArxoXNwIA85/UqsAJfQCAg4MSfe93dXWhC4OdO3cSoVcpAJg1axb79LEzLXGyF9Uxn8+ngfS1GKfv4wCRlZWVEr9Fp+lv376dGU6di+zevVvRNCXCCV3XQZVLq9jooEp5eTkhZO7cubir0Hd+nAMjMnCJNWtISAgAjBkzhq6zRrDdmD17NiGEuRgvODhYoat7/PgxelM6duwYMxz7M8OHD1eXAwxO6DrNnTt3QMpiWRQiziExMDAgws46ACxYsEChXFDozzzzjLhbB7qkLS8vjxk+Z84cYPzoKyoqimqd/s2LDaWlpXjWc889xwzH6xKZf8YNLw4q6OM8fPgw6o95NCoqiunTGatwVAmV2sSJE1NTU3F4kY04aFeE6Z9RIBDgggljY+PRo0eLnIIdaz6f397ejs7UGxsblyxZgumw99/24MED9ByIjQNl6dKl9G2Y3hZO6IOQnJwcbNOZ4MoJHEYkQi/MlDfffJN5yrVr1wg7oTc3N9MSkpKSgoHDhg0bNWpUX18fHhLp2JiZmQHAo0ePcHQfXdx0d3ejarEosgS7+MBwY43usUD42hAdHX358mX2CUqEE7qOQt1+iEM/DBGh4ABg06ZNROjdnO6yZ9OmTSK52NvbozuQlpYWDKH+DnAcJiAggAjdOVF3VI6OjgCg0BzJ+/fv0xwJIdeuXcPBzZkzZwIAlpywsDCFLkccTug6Sm9vr0hlHBISIu7YBBt0ptPxnp4e5XyQV1dXU1+kH374IU2fEHLo0KFXX32VloFFixZRS2i3h7Jw4ULq1JI96HGEgv+f2bZtG+5mZWUpcUVMOKEPHtQy/4n5PiqSIBYhkeEXIvTtHx4evmLFChz8UY6IiAh3d/fQ0FBmYFpaWlpamtJpUjihc0hlMC0s5ITOMSTghM4xJOCEzjEk4ITOMSQAQsiaNWtOnDjR35ZwcGgQIISsXbtW9m+zOTgGOlzXhWNI8C9qo4ibZiGi4QAAAABJRU5ErkJggg==)
Since this is the equation of a parabola, therefore, the path traced by a projectile is a parabola. The following are the important equations used in projectiles:
1. The time of flight (t) of a projectile on a horizontal plane is given by
![The time of flight (t) of a projectile on a horizontal plane The time of flight (t) of a projectile on a horizontal plane](data:image/png;base64,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)
2. The horizontal range (R) of a projectile is given by
![The horizontal range (R) of a projectile The horizontal range (R) of a projectile](data:image/png;base64,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)
For a given velocity of projectile, the range will be maximum when sin 2α = 1 or α=45.
3. The maximum height (H) of a projectile on a horizontal plane is given by
![The maximum height (H) of a projectile on a horizontal plane The maximum height (H) of a projectile on a horizontal plane](data:image/png;base64,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)
4. The time of flight of a projectile when it is projected from O on an upward inclined plane as shown in Fig. 1.31, is given by
![The time of flight of a projectile The time of flight of a projectile](data:image/png;base64,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)
where β is the inclination of plane OA with the horizontal.
![](data:image/png;base64,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)
When the projectile is projected on a downward inclined plane, then
![](data:image/png;base64,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)
5. The range of projectile when it is projected from O to B on an upward inclined plane, as shown in Fig: 1.31, is given by
![](data:image/png;base64,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)
For a given velocity of projectile, the range will be maximum when
![](data:image/png;base64,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)
When the projectile is projected on a downward inclined plane, then
![](data:image/png;base64,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)
Let
O = Point of projection,
u = Velocity of projection, and
α = Angle of projection with the horizontal.
Consider a point P as the position of particle, after time t seconds with x and y as co-ordinates, as shown in Fig. 1.30.
The equation of the path of a projectile or the equation of trajectory is given by
Since this is the equation of a parabola, therefore, the path traced by a projectile is a parabola. The following are the important equations used in projectiles:
1. The time of flight (t) of a projectile on a horizontal plane is given by
2. The horizontal range (R) of a projectile is given by
For a given velocity of projectile, the range will be maximum when sin 2α = 1 or α=45.
3. The maximum height (H) of a projectile on a horizontal plane is given by
4. The time of flight of a projectile when it is projected from O on an upward inclined plane as shown in Fig. 1.31, is given by
where β is the inclination of plane OA with the horizontal.
When the projectile is projected on a downward inclined plane, then
5. The range of projectile when it is projected from O to B on an upward inclined plane, as shown in Fig: 1.31, is given by
For a given velocity of projectile, the range will be maximum when
When the projectile is projected on a downward inclined plane, then
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